Sample-efficient evidence estimation of score based priors for model selection

TL;DR

This paper introduces DiME, a novel, sample-efficient estimator for model evidence in diffusion priors, enabling effective model selection and diagnosis in complex inverse problems.

Contribution

DiME leverages intermediate diffusion sampling samples to accurately estimate model evidence with minimal evaluations, improving prior selection in Bayesian inverse problems.

Findings

DiME accurately estimates model evidence compared to analytical benchmarks.

It successfully selects correct diffusion priors in ill-conditioned inverse problems.

DiME diagnoses prior misfit in real-world black hole imaging tasks.

Abstract

The choice of prior is central to solving ill-posed imaging inverse problems, making it essential to select one consistent with the measurements $y$ to avoid severe bias. In Bayesian inverse problems, this could be achieved by evaluating the model evidence $p(y \mid M)$ under different models $M$ that specify the prior and then selecting the one with the highest value. Diffusion models are the state-of-the-art approach to solving inverse problems with a data-driven prior; however, directly computing the model evidence with respect to a diffusion prior is intractable. Furthermore, most existing model evidence estimators require either many pointwise evaluations of the unnormalized prior density or an accurate clean prior score. We propose DiME, an estimator of the model evidence of a diffusion prior by integrating over the time-marginals of posterior sampling methods. Our method leverages the large amount of intermediate samples naturally obtained during the reverse diffusion sampling process to obtain an accurate estimation of the model evidence using only a handful of posterior samples (e.g., 20). We also demonstrate how to implement our estimator in tandem with recent diffusion posterior sampling methods. Empirically, our estimator matches the model evidence when it can be computed analytically, and it is able to both select the correct diffusion model prior and diagnose prior misfit under different highly ill-conditioned, non-linear inverse problems, including a real-world black hole imaging problem.